Tom Bombadil is an ancient being, that is known in many parts of Middle-earth, by many names and in different aspects. His self-description isn't all too revealing: "Old Tom Bombadil is a merry fellow; / Bright blue his jacket is, and his boots are yellow." (LOTR I, 135) The most fitting description is perhaps Goldberry's, given in answer to Frodo's question: "Tell me, if asking does not seem foolish, who is Tom Bombadil?" "He is Master of wood, water, and hill." (ibid.) Which doesn't mean, as the hobbits think, that all this belongs to him, but -vice versa- that he belongs to wood, water and hill, and there is master of himself. He is called Iarwain Ben-adar, oldest and fatherless by the Elves, Orald by the Northern Men and Forn by the Dwarves (LOTR I, 278). His fame comes from a time, when the Old Forest covered large parts of the world, from the Shire down to Isengard, and Bombadil walked all the woods. The woods are diminished, but Bombadil still doesn't care to leave what is now the Old Forest near the river Baranduin (LOTR I, 278f). His house on a hill is not far from the Withywindle, with a view on the Barrow-downs to the east; there he lives with the love of his unending life: Goldberry, the River-daughter.
Elrond thought of bringing him to the Council, but Gandalf said, that "he would not have come". Bombadil might have the power to keep the One Ring safe in his realm, as Elronds chief counsellor Erestor suggested, but he is too light-hearted for such a task (LOTR I, 278f). Sam wished for his presence in Shelob's lair - most fittingly, because She is old too and was never disturbed by the cries of elves (LOTR II, 328); but while he had helped with the Barrow-wights (LOTR I, 153-157) he "will not pass the borders" of his country, because "Tom has his house to mind, and Goldberry is waiting!" (LOTR I, 159). Borders, that -according to Gandalf- "none can see", and that Bombadil has set himself, and will not transgress (LOTR I, 279). Yet Gandalf is wrong, when he tells Frodo, that Bombadil would be "not much interested in anything that we have done or seen, unless perhaps in our visit to the Ents" (LOTR III, 275). Just as Treebeard, a "moss-gatherer" (ibid.) like Bombadil, who doesn't leave his forest, was "immensely interested in everything" the hobbits could tell him - so Tom Bombadil keeps himself informed about the events outside his realm, and knows about Aragorn and his doings just as well as about the current innkeeper of the Prancing Pony (LOTR I, 159).
Tom Bombadil made his first appearance in the "Oxford Magazine" about 1933 (Letters, 178). A 'last appearance' in a film was not to come to pass, not in the least because Tolkien himself never liked the Story-lines that involved him (Letters, 267, 272), just as he didn't like him in the broadcast (Letters, 228). Though he doesn't seem to be all-important to the story, Tolkien paid him and his appearance a great deal of importance, up to the feather on his hat, which changed from earlier to post-LOTR versions (Letters, 318f). His origin is as "the spirit of the (vanishing) Oxford and Berkshire countryside" (Letters, 26), not, as some have suggested, originally from a doll. In the parody he becomes Tim Benzedrino, and with his "mush-brained" wife Hashberry lives a happy live of everlasting celebration, serving quite funny mushrooms to Frito and the others (BOTR, 21-30). This goes along well with the Adventures of Tom Bombadil, where he talks to trolls and birds alike and lives a quite jolly life. But as Tolkien himself stated, there's more to the character than meets the eye.
I close with J. R. R. Tolkien's own words: "And even in a mythical Age there must be some enigmas, as there always are. Tom Bombadil is one (intentionally). (Letter to Naomi Mitchison; Letters, 174)
References:
LOTR: Lord of the Rings, London (Harper Collins) 1991
Letters: The Letters of J.R.R. ed. Humphrey Carpenter, Boston (Haughton Mifflin) 1981
ATB: The Adventures of Tom Bombadil; in: The Tolkien Reader, New York (Ballantine) 1966.
BOTR: Bored of the Rings, by H.N.Beard and D.C. Kenney. New York (Roc) 1993
Hypertextual Systems by FMI Publishing, 1996